UBC Theses and Dissertations

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UBC Theses and Dissertations

A survey of recent results on torsion free abelian groups Duke, Stanley Howard

Abstract

This thesis is a survey of some recent results concerning torsion free abelian groups, hereafter referred to as groups. The emphasis is on countable groups, particularly groups of finite rank. Section 1 contains the introduction and some notation used throughout this thesis. We begin in section 2 by describing the general nature of the existing characterizations for countable groups and by describing why these characterizations do not provide satisfactory systems of invariants. We include here a brief description of a classification for groups of arbitrary power. Pathologies of groups are discussed in section 3. We briefly discuss rank one groups and completely decomposable groups and then present examples to show the vast number of indecomposable groups which exist and that a group may have two different decompositions into the direct sum of indecomposable groups. Quasi-isomorphism and the ring of quasi-endomorphisms of a group are introduced in section 4 and discussed briefly. We present the theorems which establish the importance of these notions; namely that (i) quasi-decompositions of certain groups are unique up to quasi-isomorphism and (ii) the quasi-decomposition theory of certain groups is equivalent to the decomposition theory of the quasi-endomorphism ring considered as a right module over itself. Included under 'certain groups’ are the groups of finite rank. Section 5 is devoted to rank two groups. We outline the development of the quasi-isomorphism invariants for rank two groups, due to Beaumont and Pierce, and discuss some of their applications. For example, conditions, in terms of the invariants, are given for quasi-isomorphic rank two groups to be isomorphic. Type sets are reviewed in section 6. We present both necessary and sufficient conditions for sets of types to be the type sets of rank two groups and of groups of arbitrary finite rank. We devote section 7 to a brief discussion of the notion and importance of quasi-essential groups. The ideas of irreducibility and the psuedo-socle are defined in section 8. We demonstrate how these ideas affect the structure of the quasi-endomorphism ring by showing how they can be used to compete the quasi-endomorphism ring of rank two groups.

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